On optimal estimation of the mode in nonparametric deconvolution problems

摘要

This work deals with the problem of estimating the mode in nonparametric deconvolution models. First, given n i.i.d. observations from Y = X + ε, we consider estimating the mode θ of a density function of some random variable X. Second, we consider the errors-in-variables regression model, where we are interested in the mode of m(x) = E(Z|X = x), where n i.i.d. observations from (Y, Z) with Y = X + ε are given. In both cases, we assume the distribution of ε to be ordinary smooth. The mode estimator θ_n is defined via maximising over a curve estimator of the kernel type. In both deconvolution models, we obtain rates for the quadratic risk of θ_n, depending on the smoothness of the underlying curve and the degree of ill-posedness of the deconvolution problem. Further, we show that these rates are optimal, considering one-dimensional subproblems in the class of functions studied.